Question

A teacher believes that the third homework assignment is a key predictor in how well students will do on the test. Let x represent the third homework score and y the test score. A random sample of last terms students were selected and their grades are shown below. Assume scores are normally distributed.

HW3	term
13.3	59.811
21.9	87.539
9.7	53.728
25	96.283
5.4	39.174
13.2	66.092
20.9	89.729
18.5	78.985
20	86.2
15.4	73.274
25	93.25
9.7	52.257
6.4	43.984
20.2	79.762
21.8	84.258
23.1	92.911
23	87.82
11.4	45.034
14.9	71.869
18.4	76.704
15.1	60.431
15	65.15
16.8	77.208

Approximately what percentage of the variation in the test grade is accounted for by the HW3 grade in this model?

Place your answer, rounded to 1 decimal place, in the blank. Do not use any stray punctuation marks or a percentage sign. For example, 78.9 would be a legitimate entry.

Answer 1

Solution :

Here test grades depends on HW3 grades. Hence HW3 grade is independent variable and test grade (term) is dependent variable.

Hence define variables x and y as:

x: HW3 grade

y: test grade (term)

We know that percentage of variation in dependent variable explained by independent variable is measured by R_Squared value in regression analysis.

Lets perform regression analysis using R. R code :

#enter dependent variable values x=c(13.3, 21.9, 9.7, 25 , 5.4 , 13.2, 20.9, 18.5, 20, 15.4, 25, 9.7, 6.4, 20.2, 21.8, 23.1, 23, 11.4, 14.9, 18.4, 15.1,15, 16.8)

#enter independent variable values

y=c(59.811, 87.539, 53.728, 96.283, 39.174 , 66.092, 89.729, 78.985 , 86.2, 73.274, 93.25, 52.257, 43.984, 79.762 , 84.258 , 92.911 , 87.82 , 45.034 , 71.869, 76.704, 60.431 , 65.15 , 77.208

)

#fit the regression model using lm() function

model = lm( y~x)

model

#summay of model

summary(model)

Output :

Call:

lm(formula = y ~ x)

Coefficients:

(Intercept) x  

23.395 2.925  

Call:

lm(formula = y ~ x)

Residuals:

Min 1Q Median 3Q Max

-11.7024 -2.5968 0.0937 3.0273 5.2084

Coefficients:

Estimate Std. Error t value Pr(>|t|)   

(Intercept) 23.3954 2.7998 8.356 4.07e-08 ***

x 2.9247 0.1591 18.388 2.00e-14 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.245 on 21 degrees of freedom

Multiple R-squared: 0.9415, Adjusted R-squared: 0.9387

F-statistic: 338.1 on 1 and 21 DF, p-value: 1.999e-14

Conclusion :

From the regression analysis report, we have value of R squared is 0.9415

Hence percentage of variation explained in test grade accounted for by HW3 grade is 0.9415×100=94.15 i.e 94.2.

Answer : 94.2